Exploring gravitational theories beyond Horndeski

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Exploring gravitational theories beyond Horndeski

Jérôme Gleyzes, David Langlois, Federico Piazza, Filippo Vernizzi

To cite this version:

Jérôme Gleyzes, David Langlois, Federico Piazza, Filippo Vernizzi. Exploring gravitational theories beyond Horndeski. Journal of Cosmology and Astroparticle Physics, Institute of Physics (IOP), 2015, 2015 (02), pp.018. �10.1088/1475-7516/2015/02/018�. �hal-01261098�

arXiv:1408.1952v2 [astro-ph.CO] 11 Feb 2015

Exploring gravitational theories beyond Horndeski

J´erˆome Gleyzes^a,b, David Langlois^c, Federico Piazza^c,d,e and Filippo Vernizzi^a

a CEA, IPhT, 91191 Gif-sur-Yvette c´edex, France CNRS, URA-2306, 91191 Gif-sur-Yvette c´edex, France

b Universit´e Paris Sud, 15 rue George Cl´emenceau, 91405, Orsay, France

c APC, (CNRS-Universit´e Paris 7), 10 rue Alice Domon et L´eonie Duquet, 75205 Paris, France

d Aix Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France.

e Physics Department and Institute for Strings, Cosmology, and Astroparticle Physics, Columbia University, New York, NY 10027, USA

February 13, 2015

Abstract

We have recently proposed a new class of gravitational scalar-tensor theories free from Ostro- gradski instabilities, in Ref. [1]. As they generalize Horndeski theories, or “generalized” galileons, we call them G³. These theories possess a simple formulation when the time hypersurfaces are chosen to coincide with the uniform scalar field hypersurfaces. We confirm that they contain only three propagating degrees of freedom by presenting the details of the Hamiltonian formulation.

We examine the coupling between these theories and matter. Moreover, we investigate how they transform under a disformal redefinition of the metric. Remarkably, these theories are preserved by disformal transformations that depend on the scalar field gradient, which also allow to map subfamilies of G³into Horndeski theories.

Contents

1 Introduction 2

2 Galileons and Horndeski theories 4

2.1 Galileon theories . . . 4

2.2 Coupling to gravity and Horndeski theories . . . 5

3 Beyond Horndeski: G³ 6 4 Hamiltonian analysis 8 4.1 Lagrangians up to L4. . . 9

4.2 Including the Lagrangian L₅ . . . 12

4.3 Generalizations . . . 13

5 Linear theory and coupling with matter 13 5.1 Unitary gauge . . . 13

5.2 Adding matter: P(σ, Y) . . . 15

5.3 Newtonian gauge . . . 16

6 Field redefinitions 17 6.1 Disformal transformations . . . 17

6.2 Link between L₄ and Horndeski . . . 19

6.3 Link between L₅ and Horndeski . . . 20

6.4 Coupling to matter . . . 20

6.5 Equations of motion . . . 21

7 Conclusions 22

A Covariant theory 23

B Connection to the building blocks of dark energy 25

C Disformal transformation in Newtonian gauge 28

1 Introduction

The fact that current cosmological observations consistently point to a recent phase of accelerated expansion has boosted the exploration of alternative theories of gravity (seee.g.[2] for a review), that could provide a more natural explanation than simply a cosmological constant. Even if these efforts have not led to a compelling or even realistic model, these research activities have deepened our un- derstanding of gravity by highlighting the theoretical and observational constraints that alternatives to general relativity must satisfy.

Many models of modified gravity involve the presence of at least one scalar degree of freedom in addition to the two tensor degrees of freedom of general relativity. The underlying scalar field can sometimes be hidden in the explicit formulation of the theory. A typical example is f(R) theory, where the Lagrangian is written as a function of the Ricci scalar R, but which can be reformulated as a manifestly scalar-tensor theory (see e.g.[3]).

A minimal requirement on alternative theories is the absence of ghost-like instabilities within their domains of validity (see e.g. [4] on this point). According to the so-called Ostrogradski’s theorem,

such instabilities arise in theories characterized by a non-degenerate Lagrangian¹ with higher time derivatives (seee.g. [5]). The simplest example is the Lagrangian

L= 1

2q¨², (1)

which leads to fourth-order equations of motion. In the Hamiltonian formulation, an extra degree of freedom appears so that the corresponding phase space is four-dimensional, with a Hamiltonian that depends linearly on one of the momenta and is thus (kinetically) unbounded from below. In this case the extra degree of freedom is a ghost and the theory is not viable.

Not all theories containing higher-order time derivatives in the Lagrangian suffer from Ostro- gradski instabilities. In particular, this is the case for theories that lead to second-order equations of motion, such as the much studied galileon models [6], briefly reviewed in Sec.2.1. Although originally introduced in Minkowski, the galileon Lagrangians can be extended to general curved spacetimes by promoting the derivatives to covariant derivatives. However, as discussed in Sec. 2.2, maintaining second-order equations of motion with respect to spacetime derivatives requires the addition of suit- able gravitational “counterterms” [7, 8]. The largest class of these Generalized Galileons [9], or G², turns out to be equivalent to the more ancient Horndeski’s theories [10], which correspond to the most general scalar-tensor theories with second-order field equations.

Although Horndeski theories are often considered as the most general scalar-tensor theories im- mune from Ostrogradski’s instabilities, we have recently showed that this is not the case and proposed a new class of scalar-tensor theories, reviewed in Sec. 3 (see also Appendix A for the details of the calculations), that do not suffer from such instabilities [1]. Since our theory contains generalized galileons (Horndeski) as a special limit, we dubbed it “Generalized Generalized Galileons” or G³ for brevity. It turns out that our theories have the same decoupling limit as Horndeski theories, as briefly showed at the end of Sec.3.

The stability properties of G³ are most easily seen by using the ADM formalism applied to the uniform scalar field hypersurfaces (also called unitary gauge formulation). In this formulation, the scalar field does not appear explicitly as it is part of the degrees of freedom of the metric, and the action depends only on first time derivatives of the metric (the “velocities”), as generally expected from healthy theories. Indeed, the Hamiltonian analysis confirms the absence of unwanted extra degrees of freedom, and thus the absence of Ostrogradski instabilies [1]. In Sec. 4 of the present article we give more details about the derivation of the Hamiltonian and about the counting of the degrees of freedom, which depends on the number and nature (first or second class) of the constraints between canonical variables. Our analysis clearly proves that our theories contain only three degrees of freedom and do not suffer from Ostrogradski instabilities, as stated in [1].

Hints that one could go beyond Horndeski theories without encountering fatal instabilities ap- peared in our previous work [11], where we studied the most general quadratic Lagrangian for linear perturbations about a homogenous and isotropic spacetime that does not induce higher derivatives on the linear propagating scalar degree of freedom. Such a Lagrangian contains an additional term, which is absent in Horndeski theories. In Sec. 5.1 we review this analysis of linear perturbations and we extend it in Sec. 5.2 by including some matter field, detailing the analysis of [1]. For con- venience, we describe matter by means of a scalar field with non-standard kinetic term, which can be formulated in terms of a simple Lagrangian and which is characterized by a nontrivial speed of sound. We are thus able to derive a quadratic Lagrangian that includes both metric and matter perturbations in the unitary gauge. A similar calculation was presented in [12], and generalized to several matter scalar fields in [13]. We also give an equivalent treatment for perfect fluid matter by working directly with the equations of motion written in the Newtonian gauge, in Sec. 5.3. For this analysis we find it convenient to employ the notation proposed in Ref. [14], based on the effective

1A Lagrangian L(q,q,˙ q) is said to be nondegenate if¨ ∂²L/∂¨q²6= 0

approach to cosmological perturbations for dark energy, introduced in [15,16,11,17]. In Appendix Bwe review the connection between the different notations employed in these references.

Other fissures in the standard lore concerning Horndeski theories were pointed out in [18], which studied scalar-tensor theories generated by disformal relations [19]

˜

g_µν = Ω²(X, φ)g_µν+ Γ(X, φ)∂_µφ∂_νφ , (2) where X ≡ g^µν∂µφ∂νφ. In particular, it was shown that starting from an action consisting of the Einstein-Hilbert term for ˜g_µν and of a standard action forφ, one obtains equations of motion forg_µν and φthat are higher order but can be combined so that the dynamics is only second order. This is another example beyond Horndeski that is not Ostrogradski unstable. Interestingly, a very similar argument has been invoked in the context of ghost-free massive gravity in [20].

It is natural to wonder whether our theories could be formulated in a similar way, i.e. derived via a disformal transformation from a theory belonging to the Horndeski class. We discuss this issue in Sec. 6 and find that our general theory cannot be derived from Horndeski via a disformal transformation. Remarkably however, the two non-Horndeski pieces contained in our Lagrangian can beseparately derived from a Horndeski Lagrangian combined with a disformal transformation. Since the disformal transformation that we consider conserves the number of degrees of freedom, this proves that our two non-Horndeski pieces are separately equivalent to a subset of Horndeski theories. In AppendixCwe explicitly check in Newtonian gauge that the disformal metric redefinition de-mixes part of the kinetic couplings (the part containing higher derivatives) between the scalar field and the metric. In this respect, the disformal transformations considered here are analogous to the field redefinition removing higher derivatives discussed in the context of massive gravity in [20]. Since the two disformal transformations are distinct for the two non-Horndeski pieces of G³, the procedure cannot be applied to the whole Lagrangian. However, the fact that these pieces can be mapped to Horndeski provides an alternative way to show the healthy behavior of our theories. Using a disformal transformation, in Sec.6.5we provide an example of naively higher-derivative equations of motion which can be reduced to second order ones, generalizing the treatment of [18].

2 Galileons and Horndeski theories

2.1 Galileon theories

One of the most explored frameworks for infra-red modifications of gravity is the so-called galileon theory [6], which distills and generalizes the interesting features of the DGP scenario [21] and emerges in the decoupling limit of massive gravity [22].

Galileon theories can be seen as the effective theory of a Goldstone bosonφin Minkowski space, that is invariant under a generalized shift symmetry,

φ(x) → φ(x) +b_µx^µ+c, (3) for the five arbitrary parameters bµ and c. Only in Minkowski can we arbitrarily choose a constant vector fieldb^µand thus this is where galileon theories are naturally set. At lowest order in derivatives, there exists a limited number of Lagrangian terms invariant under (3), with schematic form Ln ∼ (∂φ)²(∂²φ)ⁿ⁻², where n≤5 in four dimensions. Such operators are protected by the symmetry (3) against quantum corrections [23,24].

These theories can be most naturally formulated as [6]

L^gal,1n+1 = (A^{µ1...µnν1...νn}φ_µ1φ_ν1)φ_µ2ν². . . φ_µnνⁿ, (4) where A^{µ1...µnν1...νn} is a tensor separately antisymmetric in the indices µ’s and ν’s and symmetric under the exchange{µ_i} ↔ {ν_i},e.g. A^µ1µ2ν1ν2 ∝g^µ1ν1g^µ2ν2−g^µ1ν2g^µ2ν1 (seee.g.the nice review [25]

for technical details). In the above expression and in the rest of this section, we use the shorthand notation φ_µ ≡ ∇µφ, φ_µν ≡ ∇ν∇µφ for convenience. More explicitly, the galileon Lagrangians are written as linear combinations of the five following Lagrangians:

L^gal,1₂ =X , (5)

L^gal,1₃ =Xφ−φµφ^µνφν , (6)

L^gal,1₄ =X

(φ)²−φ_µνφ^µν

−2(φ^µφ^νφ_µνφ−φ^µφ_µνφ_λφ^λν), (7) L^gal,1₅ =X

(φ)³−3(φ)φ_µνφ^µν+ 2φ_µνφ^νρφ^µ_ρ

(8)

−3

(φ)²φ_µφ^µνφ_ν−2φφ_µφ^µνφ_νρφ^ρ−φ_µνφ^µνφ_ρφ^ρλφ_λ+ 2φ_µφ^µνφ_νρφ^ρλφ_λ . In flat space there exist alternative (in fact, infinite) versions of galileon Lagrangians, equivalent up to total derivatives. A particularly compact and popular choice (called “form 3” in [25]) is

L^gal,3₂ =X , (9)

L^gal,3₃ = 3

2Xφ , (10)

L^gal,3₄ = 2X

(φ)²−φ_µνφ^µν

, (11)

L^gal,3₅ = 5 2X

(φ)³−3(φ)φ_µνφ^µν+ 2φ_µνφ^νρφ^µ_ρ

, (12)

where we have chosen the normalization factors in order to be consistent with the original expres- sions (5)-(8).

2.2 Coupling to gravity and Horndeski theories

By going from (5)-(8) to (9)-(12) we have exchanged the order of partial derivatives, which can be consistently done in flat space. But in general curved spaces, while doing so for L₄ and L₅ we have to pay a commutator proportional to the curvature. Indeed, by taking f as a general function ofX, we find that the two main blocks of terms appearing in L^gal,1₄ and L^gal,1₅ are related by, respectively,

f

(φ)²−φ_µνφ^µν

=−2f_X(φ^µφ^νφ_µνφ−φ^µφ_µνφ_λφ^λν) +f ⁽⁴⁾R^µνφ_µφ_ν + boundary terms, (13) and

f

(φ)³−3(φ)φ_µνφ^µν + 2φ_µνφ^νρφ^µ_ρ

=

−2f_X

(φ)²φ_µφ^µνφ_ν−2φφ_µφ^µνφ_νρφ^ρ−φ_µνφ^µνφ_ρφ^ρλφ_λ+ 2φ_µφ^µνφ_νρφ^ρλφ_λ

−2Xf

(4)R_µσρνφ^µφ^ρσφ^ν +⁽⁴⁾R_µνφ_σφ^µσφ^ν−⁽⁴⁾R_µνφ^µφ^νφ

+ boundary terms.

(14)

This also means that the different versions of the galileon Lagrangians, which are all equivalent in flat space, correspond to genuinely different theories once minimally coupled to gravity by trading ordinary derivatives for covariant derivatives. Of course, as realized in [7], the minimally coupled versions of galileons L₄ and L₅ bring higher (third order) derivatives into the equations of motion.

For example, by varyingX(φ)²with respect toφ, one ends up with terms containing two derivatives hitting on a Christoffel symbol, i.e., three derivatives of the metric. In order to get rid of such higher derivatives, the authors of [7] added toL^gal,1₄ andL^gal,1₅ suitable gravitational “counterterms” and thus

“re-discovered” Horndeski theories [10], which can be described by an arbitrary linear combination

of the Lagrangians

L^H₂ [G₂]≡G₂(φ, X), (15)

L^H₃ [G₃]≡G₃(φ, X)φ , (16)

L^H₄ [G₄]≡G₄(φ, X)⁽⁴⁾R−2G_4X(φ, X)(φ²−φ^µνφ_µν), (17) L^H₅ [G₅]≡G₅(φ, X)⁽⁴⁾G_µνφ^µν+ 1

3G_5X(φ, X)(φ³−3φ φ_µνφ^µν+ 2φ_µνφ^µσφ^ν_σ), (18) following the presentation given in Ref. [9].

3 Beyond Horndeski: G

As we have recently shown in [1], it turns out that it is possible to extend the Horndeski Lagrangians presented above without encountering ghost-like Ostrogradski instabilities. In order to introduce these theories, it is much simpler to use the so-called unitary gauge, where the uniform scalar field (φ= const) hypersurfaces coincide with constant-time hypersurfaces. To do so, we assume that the gradient of the scalar field,∂_µφ, is time-like. Using an ADM decomposition of the metric,

ds² =−N²dt²+h_ij(dxⁱ+Nⁱdt)(dx^j+N^jdt), (19) we write the Lagrangian density in terms of the intrinsic and extrinsic 3-d curvature tensors of the spatial slices, respectively denoted R_ij and K_ij, their traces,R ≡h^ijR_ij,K≡h^ijK_ij, as well as the lapse function N. The theories presented in [1] are then given by the action

S= Z

d⁴x√

−g(L₂+L₃+L₄+L₅), (20) with

L₂ ≡A₂(t, N), L₃ ≡A₃(t, N)K ,

L4 ≡A4(t, N) K²−KijK^ij

+B4(t, N)R , L₅ ≡A₅(t, N) K³−3KK_ijK^ij + 2K_ijK^ikK^j_k

+B₅(t, N)K^ij

R_ij−1 2h_ijR

,

(21)

where Aa and Ba (a= 2,3,4,5) are generic functions of t and N. Let us remind that, in terms of ADM variables, the extrinsic curvature reads

K_ij = 1

2N h˙_ij −D_iN_j−D_jN_i

, (22)

where D_i is the spatial covariant derivative. The combination K²−K_ijK^ij in the third line is the usual GR kinetic term. Indeed, whenB₄ =−A₄= 1/(16πG), while the other coefficients vanish, the above action corresponds to the Einstein-Hilbert action up to boundary terms, as can be easily seen upon using the Gauss-Codazzi relation (see eq. (132) in App. A). In this case the action becomes fully 4-d diff invariant and there are no propagating scalar degrees of freedom.

We now rewrite the above Lagrangians in a manifestly covariant form, i.e. in terms of φ and its spacetime derivatives. The dependence on t and N of the functions A_a and B_a will turn into a dependence onφandX≡g^µν∂µφ∂νφ, sinceφ=φ0(t) andX=−φ˙²₀(t)/N²in our ADM formulation.

We can then introduce the unit vector normal to the uniformφhypersurfaces, n_µ≡ − ∂µφ

√−X, (23)

and define the extrinsic curvature as

K_µν ≡(g^σ_µ+n^σn_µ)∇σn_ν. (24) Using this expression and K ≡ ∇µn^µ, and denoting the derivation by a lower index, e.g. A_2X ≡

∂A₂/∂X, the above Lagrangians can be rewritten, after lengthy but straightforward manipulations explicitly given in App.A, as [1]

L₂ =L^H₂ [A₂], (25)

L₃ =L^H₃ [C₃+ 2XC_3X] +L^H₂ [XC_3φ], (26)

L₄ =L^H₄ [B₄] +L^H₃ [C₄+ 2XC_4X] +L^H₂ [XC_4φ]−B₄+A₄−2XB_4X

X² L^gal,1₄ , (27)

L5 =L^H₅ [G5] +L^H₄ [C5] +L^H₃ [D5+ 2XD5X] +L^H₂ [XD_5φ] +XB5X + 3A5

3(−X)^5/2 L^gal,1₅ , (28) where Aa and Ba are now functions of φ and X, Aa =Aa(φ, X), Ba = Ba(φ, X), and C3, C4, C5, D₅ and G₅ are defined as

C₃ ≡ 1 2

Z

A₃(−X)^−3/2dX , C₄ ≡ −

Z

B_4φ(−X)^−1/2dX , C₅ ≡ −1

4X Z

B_5φ(−X)^−3/2dX , D₅ ≡ −

Z

C_5φ(−X)^−1/2dX , G5 ≡ −

Z

B5X(−X)^−1/2dX .

(29)

IfA4 and A5 are related to B4 and B5 by

A₄ =−B₄+ 2XB_4X , A₅=−XB_5X/3, (30)

the last terms of both eqs. (27) and (28) vanish. In this case one is left only with the Horndeski Lagrangians, which manifestly shows that eqs. (25)–(28) (and thus action (20)) contain Horndeski theories. In general, the functions A₄ and A₅ are completely free, which means that our theories contain two additional free functions with respect to the Horndeski ones.

It is straighforward to see that the minimally coupled versions of the original galileons proposed in [6], (5)–(8), are contained in eqs. (25)–(28) by the choice of functionsB₄ = 0,B₅ = 0,A₂ =X,A₃ = 3X/2,A₄ =−X²andA₅ = (−X)^5/2. As a corollary,L^gal,1₄ andL^gal,1₅ are already healthy without the need of additional gravitational counterterms. In other words, the straightforward covariantization of galileons, i.e. substituting ordinary derivatives with covariant derivatives, is a viable covariantization.

It should be noted, however, that galileon symmetry remains broken by terms proportional to the curvature, regardless of the chosen covariantization procedure.

Finally, before concluding this section, let us briefly comment on the decoupling limit of eqs. (25)–

(28). In Ref. [26], the decoupling limit of Horndeski theories has been studied by expanding the metric g_µν around Minkowski and the scalar field φ around a constant background value. In doing so, the following scaling of the functionsGa(φ, X) introduced in eqs. (15)–(18) was assumed [27],

G₂∼Λ³₃M_Pl, G₃ ∼M_Pl, G₄ ∼M_Pl² , G₅ ∼Λ⁻³₃ M_Pl² , (31) where Λ₃is a mass scale which may be associated to the current accelerated expansion of the universe (in which case Λ³₃ ∼M_PlH₀²) andM_Plis the Planck mass. The decoupling limit is defined asM_Pl→ ∞

while Λ₃ remains constant. It is easy to see that taking this limit in eqs. (25)–(28) leads to the same decoupling limit found in [26] for Horndeski, but with different dimensionless parameters. This is clearly the case for eqs. (25) and (26), because they are equivalent to the Horndeski LagrangiansL^H₂ and L^H₃ . Equations (27) and (28) contain non-Horndeski pieces, respectively L^gal,1₄ and L^gal,1₅ . By expanding these terms in scalar field and metric perturbations, the only contributions that do not vanish in the decoupling limit are galileons, i.e.,

−B4+A4−2XB4X

X² L^gal,1₄ ∼Λ⁻⁶₃ L^gal,1₄ , XB5X + 3A5

3(−X)^5/2 L^gal,1₅ ∼Λ⁻⁹₃ L^gal,1₅ , (32) where the functions (B₄+A₄−2XB_4X)/X² and (XB_5X/3 +A₅)/(−X)^5/2 are evaluated on the background. In conclusion, operators leading to higher-derivative equations of motion in eqs. (27) and (28) are also higher order in the decoupling limit.

4 Hamiltonian analysis

As discussed in the introduction, theories that contain higher-order time derivatives often lead to lethal Ostrogradski instabilities. The presence of higher derivatives manifests itself in the form of extra degrees of freedom that behave like ghosts (i.e. negative energy states). For instance, the dynamics of a system with a nondegenerate Lagrangian of the form L(q,q,˙ q) is described by a 4-¨ dimensional phase space, corresponding to two degrees of freedom, one of which behaves like a ghost (see e.g.[5]).

In the ADM formulation, our Lagrangian (21) depends on the dynamical quantitiesh_ij and their

“velocities” Kij: in this sense, it is already evident that the Lagrangian does not contain higher- order time derivatives and that Ostrogradski instabilities should not be there. In order to confirm this intuition, we now perform the Hamiltonian analysis for the Lagrangian (21) and show that the number of degrees of freedom remains three—i.e. two tensor modes and one scalar mode, thus excluding the appearance of dangerous extra degrees of freedom. The present analysis details that of [1] and confirms its conclusions.

The phase space of our theory is described by the variables h_ij, N, Nⁱ and their conjugate momenta, given respectively by

π^ij ≡ ∂L

∂h˙ij

=

√h 2

hA₃+ 2A₄K+ 3A₅(K²−K_lmK^lm) h^ij

−2(A₄+ 3A₅K)K^ij+ 6A₅K_lⁱK^lj+B₅

R^ij−1 2Rh^ij

,

(33)

and

π_N ≡ ∂L

∂N˙ = 0 π_i≡ ∂L

∂N˙ⁱ = 0. (34)

The absence of time derivatives of the lapse N and the shift Nⁱ in the action implies that their conjugate momenta automatically vanish. The relations πN = 0 and πi = 0 can thus be seen as restrictions of the initial 20-dimensional phase space, corresponding to so-called primary constraints.

So far, the situation is quite similar to that of pure general relativity.

The canonical Hamiltonian is then obtained via the Legendre transform of the Lagrangian, H ≡

Z d³~xh

π^ijh˙_ij − Li

. (35)

The Hamiltonian is expressed in terms of the canonical variables, which means that, in principle, one must invert the relation in (33) to obtain ˙hij as a function ofπ^ij. Because of the presence of primary

constraints, the time evolution is governed by the extended Hamiltonian, H˜ =H+

Z d³~x

λ_Nπ_N +λⁱπ_i

, (36)

where λ_N and λ_i play the role of Lagrange multipliers. For any function F defined on the phase space, its time evolution is given by

d

dtF = ∂F

∂t +

F,H˜ . (37)

The Poisson bracket in the above formula is defined, as usual, by the expression {F, G} ≡X

A

Z d³~x

δF δφ^A(~x)

δG

δπ_A(~x)− δF δπ_A(~x)

δG δφ^A(~x)

, (38)

where we use the collective notation φ^A= (h_ij, N, Nⁱ) and π_A= (π^ij, π_N, π_i).

4.1 Lagrangians up to L₄

It is straightforward to apply the procedure outlined above to our Lagrangians up toL₄, because the expression (33) for π^ij is linear in K_ij and can be easily inverted. IncludingL₅ is more involved, as (33) is quadratic inK_ij and we briefly discuss the procedure in the next subsection.

Therefore, assuming thatL₅ is absent, i.e. A₅=B₅ = 0, one can immediately invert (33) to find Kij =− 1

A₄√ h

πij− 1

2πhij

− A₃

4A₄hij. (39)

Using (22), it is then straightforward to express ˙h_ij as a function of π_ij and to substitute the result in (35). Using integrations by parts to get rid of the derivatives of the shift, one finds that the Hamiltonian can be written in the form

H = Z

d³~x

NH0(N) +NⁱHi

, (40) with

H⁰ ≡ − 1

√hA₄

πijπ^ij−1 2π²

− A3

2A₄π+√ h

3A32

8A₄ −A2

−√

h B4R , (41)

Hi ≡ −2D_jπ^j_i. (42)

As mentioned in the previous section, by specializing the above expressions to the case B₄ =−A₄ = 1/(16πG) and A₂ = A₃ = 0 one recovers the usual general relativity Hamiltonian. In the general case, however, the A_a and B_a are functions of N, so that H0 now depends on N, in contrast with general relativity. This difference plays a crucial role, as we will see below.

Let us now consider the time evolution of the primary constraints. Imposing that they are conserved in time leads to the so-called secondary constraints. For the first constraint, π_N ≈0, one finds

˙ π_N =

π_N,H˜ ≈

π_N, H =− ∂

∂N (NH0) , (43)

where the symbol≈denotes equality in a “weak” sense, i.e. restricted to the constrained phase space.

Thus, the above equation yields the secondary constraint, H˜0 ≡ H0+N∂H0

∂N ≈0. (44)

Note that, in general relativity, H0 is independent of N, thus leading to the familiar Hamiltonian constraint ˜H0=H0 ≈0. Similarly, using

˙ π_i=

π_i,H˜ ≈

π_i, H =−Hi, (45)

the conservation in time of the three primary constraints π_i≈0 gives the secondary constraints

Hi ≈0. (46)

These constraints are exactly the same as in pure general relativity, where they are associated with the invariance under spatial diffeomorphims.

Let us now compute the Poisson brackets of the constraints. We start with the constraints Hi, for which the treatment is very similar to general relativity. It is convenient to introduce the

“momentum” function

Mf ≡ Z

d³~x fⁱ(~x)Hi(~x), (47) where thefⁱare three arbitrary functions of space. By reproducing the general relativity calculations (see e.g.the appendix of [28]), one finds

{Mf,Mg}=Mh, hⁱ ≡f^kD_kgⁱ−g^kD_kfⁱ. (48) It is also straightforward to check that

{Mf,Tg}=− Z

d³~x g D_i(Tfⁱ) = Z

d³~xTfⁱD_ig , (49) with

Tg ≡ Z

d³~x g(~x)T(~x), (50) wheregis an arbitrary function of space andT is any combination of the Hamiltonian that depends on π^ij and h_ij, butnot on N. SoT can be any of the following expressions,

T1= 1

√h π_ijπ^ij−1 2π²

, T2=π , T3 =√

h , T4=√

hR , (51)

or any linear combination of these with coefficients independent of N. In particular, (49) implies that in general relativity, where the constraint H0 does not depend onN, the Poisson bracket ofMf

withH0 weakly vanishes.

If the combination T is now multiplied by a function ofN,

T˜ =F(N)T , (52) one immediately deduces from (49) that

Mf,T˜g =− Z

d³~x gFD_i(Tfⁱ), (53) and ˜T cannot appear after integration by parts. However, by introducing the slightly modified constraints²

H˜i ≡ Hi+π_N∂_iN, (54)

2Note that its form is similar to the total momentum constraint that would arise in general relativity with a scalar field.

one obtains

M˜f,T˜g =

Mf,T˜g − Z

d³~x g ∂F

∂NTfⁱD_iN =− Z

d³~x g D_i( ˜Tfⁱ) = Z

d³~xT˜fⁱD_ig , (55) where now ˜T appears explicitly.

This treatment also applies to any linear combination of ˜T terms. In particular, it applies toH⁰, since this is given by a linear combination ofTawith coefficients that depend on time and N, and as a consequence it applies to ˜H0 defined in eq. (44). Thus, from the above analysis one concludes that the Poisson brackets of the constraints ˜Hⁱ with ˜H⁰ vanish weakly, i.e.

H˜i,H˜0 ≈0. (56) Using eq. (48) and the fact that Hi does not depend on N, π_N, Nⁱ or π_i, it is also immediate to verify that

H˜i,H˜j ≈0, H˜i, π_N ≈0, H˜i, π_j ≈0. (57) Therefore, the Poisson brackets of the three constraints ˜Hi with all the other constraints vanish weakly. The same is true for the three primary constraintsπ_i≈0. Consequently, these six constraints, associated with the 3-dimensional diffeomorphism invariance, arefirst-class constraints.

The remaining constraints, ˜H⁰ and πN ≈0, satisfy the relations π_N(x), π_N(y) = 0, H˜0, π_N = ∂H˜0

∂N = 2∂H0

∂N +∂²H0

∂N² . (58)

Provided that the derivative of ˜H0 with respect to N does not vanish, this shows that these two constraints are of the second-class type, in contrast with general relativity.

It is also useful to check that no additional constraint arises from the time evolution of the secondary constraints. Indeed, since

d

dtH˜0 = ∂H˜0

∂t +H˜0, H +λ_N∂H˜0

∂N , (59)

imposing the conservation of ˜H0 simply fixes the Lagrange multiplier λ_N without generating any new constraint, provided ∂H0/∂N does not vanish, which is assumed here. As for the momentum constraints, we simply have

d

dtH˜i=H˜i, H ≈0, (60) because the brackets of ˜Hi with all the elements inH vanish weakly, according to (55) and the first relation in (57).

In conclusion, we find that the dynamical system is, in general, characterized by a 20-dimensional phase space with six first-class constraints and two second-class constraints. Each first-class con- straint eliminates two canonical variables and each second-class constraint eliminates one canonical variable. In total, 14 canonical variables can be eliminated, which corresponds to a 6-dimensional physical phase space, i.e. three degrees of freedom. The difference with general relativity, where all eight constraints are first-class thus leaving only two physical degrees of freedom, is due to the pres- ence of a preferred slicing defined by the scalar field, which breaks the full spacetime diffeomorphism invariance.

Let us briefly discuss a special case where the second Poisson bracket in (58) vanishes weakly, which happens when the whole N dependence factorizes inH0. Let us illustrate this case by consid- ering the Lagrangian L₄ with

B₄=− 1

A₄ . (61)

In this case

H0=B₄h 1 2√

h 2π_ijπ^ij −π²

−√ hRi

(62) and

H˜0=

B₄+∂B₄

∂N

h 1 2√

h 2π_ijπ^ij −π²

−√ hRi

. (63)

One then notices that the system is equivalent to general relativity, up to the redefinition of a new lapse function ˜N ≡N B₄.

Finally, let us make a few considerations on the restriction to the unitary gauge which is at the basis of the Hamiltonian analysis of this section. An explicit Hamiltonian analysis without fixing unitary gauge seems to be a very tedious task in view of the complicated expressions of our theories in the covariant form, eqs. (25)-(28). Indeed, resorting to the unitary gauge has the huge advantage to hide the scalar degree of freedom in the metric and to enormously simplify the analysis. Thus, the full Hamiltonian treatment in an arbitrary gauge is beyond the scope of the present work. Fortunately, in Sec.6we present a completely different approach, which shows that the higher-order time derivatives in the equations of motion can be eliminated by using constraints that follow from these equations.

This other approach is valid in any gauge and it confirms that no additional degree of freedom is necessary to describe higher-order time derivatives.

4.2 Including the Lagrangian L₅

The inclusion of L₅ makes the Hamiltonian analysis more involved, the main subtlety in this case being inverting eq. (33) in order to obtain Kij as a function ofπ^ij. However, this technical difficulty does not impair the basic counting of degrees of freedom, which is the main target of our Hamiltonian analysis.

In the case when only A5 is considered, from the last line of (21) we obtain π^ij = 3√

hA₅ 2

h

(K²−K_mnK^mn)h^ij + 2(Kⁱ_lK^lj−KK^ij)i

. (64)

Inverting the above equation is technically more involved and because K_ij is essentially a “square root” of πij there is generally more than one branches of solutions. However, the inversion problem is well-defined locally around some non-singular chosen value of K_ij. It is worth mentioning how the problem can be tackled in practice with a systematic series expansion around, for instance, a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) configuration,

(K₀)_i^j =Hδ^j_i, (π₀)_i^j = 3√

hA₅H²δ^j_i . (65) We can then fix whatever value of the conjugate momentum through the new “shifted” variable ˆπ_i^j,

π_i^j ≡(π₀)_i^j +3 2

√hA₅πˆ_i^j, (66)

write a formal power expansion for K_i^j,

K_i^j = (K₀)_i^j + (K₁)_i^j+ (K₂)_i^j+. . . (67) and solve (64) order by order. By doing this, we obtain the recursive relations

(K₁)_i^j = − 1 2H

ˆ π_i^j −πˆ

2δ^j_i

, (68)

(K₂)_i^j = 1 4H

h

(K₁)²−(K₁)_mⁿ(K₁)_n^m

δ^j_i + 4

(K₁)_i^l(K₁)_l^j−(K₁)(K₁)_i^ji

, . . . , (69)

where (K_a)≡(K_a)_iⁱ.

A completely analogous procedure applies to other cases, such as when the full battery of terms is present, as in eq. (33). In this case, the easily invertible part (L₂-L₄) can be used as the zeroth order piece and one can make a formal Taylor expansion in A₅.

4.3 Generalizations

Although we have focused our discussion on a specific class of theories, which represent a natural extension of Horndeski theories from the ADM point of view, similar conclusions can be drawn for a much wider class of models. Essentially, the basic ingredients that lead us to exclude the presence of unwanted additional degrees of freedom can be formulated in unitary gauge as

1. unbroken spatial diffeomorphism (producing three first-class momentum constraints as in gen- eral relativity);

2. absence of time derivatives of the lapse function N (which makes the Hamiltonian constraint an algebraic equation for N);

3. absence of time derivatives of the extrinsic curvatureK_ij (which prevents that the Lagrangian depends on the “accelerations”, i.e. the second time derivatives of h_ij).

Such an approach has already been used in the past to study, for instance, the behavior of specific models of Horava’s gravity [29]. In analogy with Horava’s gravity, one could consider various combi- nations of the intrinsic curvature tensor and its spatial derivatives, as well as various combinations of the extrinsic curvature tensor, as recently discussed in [30]. Note, however, that these theories do not generically have the same decoupling limit as Horndeski, as it is the case for G³ theories (see discussion at the end of Sec. 3).

5 Linear theory and coupling with matter

The Hamiltonian analysis excludes the presence of extra degrees of freedom. However, one still needs to check that the remaining scalar and tensor degrees of freedom are not themselves ghosts. In this section we compute the quadratic action for the perturbations of the propagating degrees of freedom and derive the conditions for which the kinetic terms have the right signs. We then add matter fields minimally coupled to gravity and study the phenomenology on small scales. We first perform this analysis in unitary gauge and then in Newtonian gauge.

5.1 Unitary gauge

Let us expand action (20) around a spatially flat FLRW metric following the general procedure developed in [11,31] (see also [32]). We use theζ-gauge and write the spatial metric as

h_ij =a²(t)e^2ζ(δ_ij +γ_ij), γ_ii= 0 =∂_iγ_ij , (70) and we split the shift as

Nⁱ=∂_iψ+N_Vⁱ , ∂_iN_Vⁱ = 0. (71) Moreover, it is convenient to express the dependence of the second-order action on the function A_a and B_a introduced in the Lagrangians (21) in terms of the following functions evaluated on the

background,³

M² ≡ −2(A₄+ 3HA₅),

α_K ≡ −2A^′₂+A^′′₂ + 3H(2A^′₃+A^′′₃) + 6H²(2A^′₄+A^′′₄) + 6H³(2A^′₅+A^′′₅)

2H²(A₄+ 3HA₅) ,

α_B ≡ −A^′₃+ 4HA^′₄+ 6H²A^′₅ 4H(A₄+ 3HA₅) , αT ≡ −B4+ ˙B5/2

A₄+ 3HA₅ −1, α_H ≡ −B₄+B₄^′ −HB₅^′/2

A₄+ 3HA₅ −1,

(72)

where a prime denotes a derivative with respect to N and a dot a derivative with respect to t. We discuss in AppendixBhow these functions are related to the general formalism of Ref. [11].

Higher (spatial) derivative terms proportional to (∂²ψ)², which are contained in quadratic prod- ucts of the extrinsic curvature, cancel from the action up to a total derivative because of the particular combinations in which these products appear in eq. (21). By varying the quadratic action with respect to Nⁱ, one obtains the momentum constraints, whose solution is N_Vⁱ = 0 and

N = 1 + 1 1 +α_B

ζ˙

H . (73)

After substitution of this equation into the quadratic action, all the terms containing ψ drop out, up to total derivatives [31]. For this reason, we do not need the Hamiltonian constraint, obtained by varying the action with respect to N, to solve forψ. After some manipulations the quadratic action becomes [11,31,14]

S⁽²⁾= 1 2

Z

d⁴x a³

Lζ˙ζ˙ζ˙²+L∂ζ∂ζ

(∂_iζ)² a² +M²

4 γ˙_ij² −M²

4 (1 +α_T)(∂_kγ_ij)² a²

, (74) where

Lζ˙ζ˙≡M²α_K+ 6α²_B

(1 +α_B)² , (75)

L∂ζ∂ζ ≡2M²(1 +α_T)− 2 a

d dt

aM²(1 +α_H) H(1 +α_B)

. (76)

As expected from the previous Hamiltonian analysis, the quadratic Lagrangian (74) does not contain higher-order time derivatives. As a consequence of the particular combination of extrinsic curvature in eq. (21), neither does it contain higher space derivatives.

The condition required to ensure that the propagating degrees of freedom are not ghost-like is that their time kinetic terms are positive, Lζ˙ζ˙>0 and M² >0. Moreover, gradient instabilities are avoided when the speed of sound of the scalar and tensor propagating degrees of freedom,

c²_s ≡ −L∂ζ∂ζ

Lζ˙ζ˙

, c²_T ≡1 +α_T , (77)

are also positive,c²_s >0 and c²_T >0.

3The first four functions in eq. (72) have been introduced by Bellini and Sawicki in Ref. [14], where they consider linear perturbations in Horndeski theories, with the differenceα^hereB =−α^thereB /2, which simplifies further the equations.

In particular, M², α^K, α^B andα^T respectively parameterize the effective Planck mass, a modification of the scalar kinetic term [34,35], a kinetic mixing between the scalar and the metric (the so-called braiding) [36,37,38,39] and a tensor speed excess. As stressed in such a reference and also shown in AppendixB, these functions are just a convenient basis of the parameters previously introduced in the context of the so-called Effective Field Theory of Dark Energy in Refs. [15,16,11,17] (see [31,33] for reviews). Here we adopt this parameterization because it simplifies the notation.

We also introduce a new function,α^H, which parametrizes the deviation from Horndeski theories [11,1].

5.2 Adding matter: P(σ, Y)

To study our theories in the presence of matter fields minimally coupled to gravity, we add to action (20) a k-essence type action describing a matter scalar field σ (not to be confused with the dark energy field φ),

S_m= Z

d⁴x√

−g P(Y, σ), Y ≡g^µν∂_µσ∂_νσ , (78) with sound speed c²_m≡P_Y/(P_Y −2 ˙σ₀²P_{Y Y}).

We can then expand at second order these actions and repeat the procedure discussed earlier. To describe matter fluctuations it is convenient to use the gauge-invariant variableQ_σ ≡δσ−( ˙σ₀/H)ζ. After substitution of the momentum constraints, the final action expressed in terms of ζ and Q_σ reads

S⁽²⁾ = Z

d⁴xa³ 1

2

L˜ζ˙ζ˙ζ˙²+ ˜L∂ζ∂ζ

(∂_iζ)² a²

−P_Y c²_m

Q˙²_σ−c²_m(∂_iQ_σ)² a²

− 2 ˙σ₀P_Y Hc²_m(1 +αB)

α_Bζ˙Q˙_σ−c²_m(α_B−α_H)∂_iζ∂_iQ_σ a²

+m²_ζζ²+m²_σQ²_σ+m²_cζQσ+λζQ˙ σ

, (79)

with the new coefficients for the kinetic and gradient terms ofζ L˜ζ˙ζ˙=Lζ˙ζ˙+ρ_m+p_m

H²c²_m

α_B 1 +α_B

2

, (80)

L˜∂ζ∂ζ =L∂ζ∂ζ−ρ_m+p_m H²

1−2(1 +α_H) 1 +α_B

, (81)

where we have used 2 ˙σ²₀PY = −(ρm +pm). The second line contains two derivative couplings between ζ and Q_σ while the third line contains non-derivative terms, which are irrelevant for the present discussion.

The kinetic matrix for (ζ, Qσ) reads M= 1

2

L˜_ζ^˙_ζ^˙ω²+ ˜L∂ζ∂ζk² A

α_Bω²−c²_m(α_B−α_H)k² A

α_Bω²−c²_m(α_B−α_H)k²

−2P_Yc⁻²_m (ω²−c²_mk²)

, A=− 2 ˙σ₀P_Y Hc²_m(1 +αB) .

(82) Requiring that its determinant vanishes yields the dispersion relation

(ω²−c²_mk²)(ω²−c˜²_sk²) = (c²_s−c˜²_s)

α_H 1 +α_H

2

ω²k², (83) with

˜

c²_s ≡c²_s−ρ_m+p_m H²M²

(1 +α_H)²

α_K+ 6α²_B . (84)

From this equation one derives the two dispersion relations ω² = c²_±k². For Horndeski theories (α_H = 0), the matter sound speed is unchanged, despite the presence of couplings in the action between the time and space derivative ofζ and Q_σ, i.e. the non- vanishing of the non-diagonal terms in the kinetic matrix. Indeed, these couplings are precisely proportional to ω²−c²_mk² and give the standard dispersion relation for matter. However, this is no longer true with our non-Horndeski extensions, where α_H 6= 0.